Math 210 - Dr. Miller - Summary #5: Definitions of Arithmetic Operations
These definitions encapsulate what it means to add, subtract, multiply, or divide, as opposed
to when to do it (what models tell us) or how to do it (which we’ll study soon). Because
the definitions strive to capture the essence of these concepts beyond what simple models
or techniques can do, they may seem very abstract to you. The key idea is to realize that
children learn what it means to add, subtract, etc., from two distinct perspectives:
1. what it means in terms of actual objects (set-based), and
2. what it means in terms of addition (constructivism - building new knowledge from old)
Addition has only one definition, and it is based on sets:
1. Set-based definition: To add numbers a + b means we take disjoint sets A and B having
a and b elements, respectively, and combine these sets by taking the union. Concisely,
a + b = n(A ∪ B),
or, children count how many objects are in a union when they add.
Subtraction has definitions from both of the perspectives above:
1. Set-based definition: To subtract numbers c−d means we take a set C having c elements
in it and pick a subset D from it having d elements. We then remove this subset using
the concept of set difference. Concisely,
c − d = n(C \ D),
or, children count how many objects are left in the set difference when they subtract.
2. Constructivist definition: Subtracting c − d to get an answer e means that e + d must
give c back again. Concisely,
c−d =e
means
e + d = c,
or, the answer must “check out” using the already known operation of addition.
Multiplication has definitions from both of the perspectives above:
1. Set-based definition: To multiply numbers f · g means we take sets F and G having f
and g elements in them, respectively, and “mix and match” them using the Cartesian
product. Concisely,
f · g = n(F × G),
or, children count how many matchings are possible when they multiply.
2. Constructivist definition: Multiplying f · g means we should add g + g + · · · + g for a
total of f addends,
or, multiplication is repeated use of the already known operation of addition.
Division has definitions from both of the perspectives above:
1. Set-based definition: To divide numbers h ÷ k means we take a set having h elements
and separate it into subsets each having k elements,
or, children count how many subsets are made when they divide.
2. Constructivist definition: Dividing h ÷ k to get an answer m means that m · k must
give h back again. Concisely,
h÷k =m
means
m · k = h,
or, the answer must “check out” using the already known operation of multiplication.
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